where \(e \colon \mathsf {c}_{\mathbbe {\Delta }}\circ \operatorname{ev}_{[0]} \rightarrow 1_{s{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}}\)  is the evaluation natural transformation of the adjunction \(\mathsf {c}_{\mathbbe {\Delta }}\dashv \operatorname{ev}_{[0]}\) .
Further, \(q \colon Q^{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\overset{\sim }{\longrightarrow }1_{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  is a cofibrant replacement functor in \({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  (which we can take to be the identity, since \(({{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }})_c = {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\) ).
The first morphism is a weak equivalence since \(\mathsf {c}_{\mathbbe {\Delta }}\)  preserves weak equivalences, and the second morphism is a weak equivalence as a consequence of the fibrancy of \(X\) ; compare to the proof of Theorem REF .
This proves the claim by [1]}.
